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RECURRENCE RELATIONS FOR NÖRLUND NUMBERS AND
BERNOULLI NUMBERS OF THE SECOND KIND
TAKASHI AGOH AND KARL DILCHER
Abstract. Known recurrence relations for the Nörlund numbers and the Bernoulli numbers
of the second kind are generalized to infinite classes of recurrence relations. The Stirling
numbers of the first kind occur in all these recurrences. Some of the relations are of arbitrary
length and contain explicit formulas as special cases. A convolution formula for Stirling
numbers of the first kind is also obtained.
1. Introduction
A large number of recurrence relations of various different types are known for the Bernoulli
numbers; see, for instance, the introduction of [2] for a historical perspective and for references. In this paper we deal with recurrence relations for two related sequences of rational
numbers, namely the Nörlund numbers and the Bernoulli numbers of the second kind.
To define the Nörlund numbers, we first consider the Bernoulli numbers of higher order,
which were studied by Nörlund [9] in connection with deeper investigations in the calculus
of finite differences. The Bernoulli numbers of order r can be defined by the generating
function
µ
¶r X
∞
n
x
(r) x
=
B
(|x| < 2π);
(1.1)
n
ex − 1
n!
n=0
(1)
[9, p. 145]. Then Bn = Bn , the ordinary Bernoulli numbers. Although usually the order r
(r)
is assumed to be a positive integer, it can be seen that Bn is a polynomial in r of degree n,
so that r can actually be extended to any real or complex value. We will not be concerned
with this, but we remark that Bernoulli numbers of negative orders are just Stirling numbers
of the second kind up to a binomial factor.
(n)
We now proceed to the sequence of numbers Bn , called Nörlund numbers by Howard [5],
i.e., higher-order Bernoulli numbers with the order equal to the index. Nörlund found the
exponential generating function
∞
X
xn
x
=
Bn(n)
(|x| < 1),
(1.2)
(1 + x) log(1 + x) n=0
n!
and the simple recurrence relation,
n
X
j=0
(j)
(−1)j Bj
= 1,
n − j + 1 j!
(0)
B0 = 1
(1.3)
The first author was supported in part by a grant of the Ministry of Education, Science and Culture of
Japan. The second author was supported in part by the Natural Sciences and Engineering Research Council
of Canada and by a Government of Japan fund for visiting faculty.
4
VOLUME 48, NUMBER 1
RECURRENCE RELATIONS FOR NÖRLUND AND BERNOULLI NUMBERS
[9, p. 150]. Nörlund also listed the first few values 1, − 12 , 56 , − 49 , 251
, − 475
, and Howard
30
12
[5] derived numerous divisibility and congruence properties. Among the uses of Nörlund
numbers are the remarkable expansions [9, p. 244],
¡
log 1 +
1
x
¢
=
∞
X
n=0
(n)
(−1)n Bn
,
(x + 1)(x + 2) . . . (x + n)
∞
1 X (−1)n Bn
1
Ψ(x) = log x − +
,
x n=1
n
(x + 1)(x + 2) . . . (x + n)
(n)
where Ψ(x) = Γ0 (x)/Γ(x) is Euler’s Ψ-function, the logarithmic derivative of the Gamma
function. Both expansions are valid for <(x) > 0.
The Bernoulli numbers of the second kind were introduced and studied by Jordan [6,
p. 265 ff.], also in connection with the calculus of finite differences. These numbers bn can
be defined by way of the generating function
∞
X
x
=
b n xn
log(x + 1) n=0
(|x| < 1);
(1.4)
[6, p. 279], or [5]. If we compare (1.4) with (1.2), as was done in [5], we immediately obtain
the relations
n
X
1
1
(n−1)
(n)
Bn = n!
(−1)n−j bj ,
bn = Bn(n) +
Bn−1
(n ≥ 1).
(1.5)
n!
(n − 1)!
j=0
The Bernoulli numbers of the second kind satisfy the simple recurrence relation
n
X
j=0
(−1)j
bj = 0
n−j+1
(n ≥ 1),
(1.6)
with b0 = 1 [6, p. 266], but note the small misprint there), and the first few values can
1
1
19
3
be easily computed as 1, 12 , − 12
, 24
, − 720
, 160
. One application of the numbers bn lies
in the integration of functions expanded in Newton series. This leads, for example, to the
interesting identities [6, pp. 277, 280]
∞
X
|bn |
1 − log 2 =
,
n+1
n=1
γ=
∞
X
|bn |
n=1
n
,
where γ is the Euler constant.
It is the purpose of this paper to extend or generalize the recurrence relations (1.3) and
(1.6). In both cases the Stirling numbers of the first kind play an important role; we summarize some of their properties in Section 2. In Section 3 we deal with the Nörlund numbers,
and in Section 4 with the Bernoulli numbers of the second kind. The final section contains
some additional remarks, including a convolution formula for Stirling numbers of the first
kind and relations with the Cauchy numbers.
2. Stirling Numbers of the First Kind
The Stirling numbers of the first and the second kind belong to the most basic objects
in combinatorics, with important applications in other areas of mathematics. In this paper
FEBRUARY 2010
5
THE FIBONACCI QUARTERLY
only the Stirling numbers of the first kind , s(n, k), will occur. They can be defined by the
generating function
n
X
x(x − 1) . . . (x − n + 1) =
s(n, k)xk .
(2.1)
k=0
This means that they are the coefficients connecting the two most fundamental bases of the
vector space of single-variable polynomials (while the inverse transformation between these
two bases is given by the Stirling numbers of the second kind).
For a combinatorial interpretation see [4]. Among their many properties the following will
be used in this paper:
s(0, 0) = s(n, n) = 1,
s(n, 0) = 0 for n ≥ 1,
s(n, 1) = (−1)n−1 (n − 1)!,
(2.3)
n
s(n, 2) = (−1) (n − 1)!Hn−1 ,
where Hk = 1 +
1
2
+ ... +
1
k
(2.2)
(2.4)
is the kth harmonic number;
(n − 1)n(3n + 2)(n + 1)
,
24
µ ¶
n
,
s(n, n − 1) = −
2
s(n, n − 2) =
(2.5)
(2.6)
in addition to s(n, k) = 0 for k < 0 and for k > n. The most basic recurrence relation is
s(n + 1, k) = s(n, k − 1) − n s(n, k),
1 ≤ k ≤ n.
(2.7)
While the generating function (2.1) has a fixed n and index of summation k, it is the other
way for the (exponential) generating function
∞
¡
¢k X
xn
k!s(n, k)
log(x + 1) =
n!
n=k
(|x| < 1),
(2.8)
which will be essential in the remainder of this paper.
These and numerous other properties can be found, e.g., in the books [1, Ch. 24], [3, 4],
or in the on-line resources [11] or [10, A008277]. Although there are some advantages to the
bracket notation used in [4] (see also [7]) we use here the main competing notation s(n, k).
3. Nörlund Numbers
It is the purpose of this brief section to derive a general recurrence relation for the Nörlund
(n)
numbers Bn . The Stirling numbers of the first kind enter via their exponential generating
function (2.8), which gives
∞
∞
¡
¢k X
xn X
xn−1
(1 + x1 ) log(x + 1) =
k!s(n, k) +
k!s(n, k)
n! n=k
n!
n=k
=
∞
X
n=k−1
6
¡
k! s(n, k) +
1
s(n
n+1
¢ xn
+ 1, k)
.
n!
VOLUME 48, NUMBER 1
RECURRENCE RELATIONS FOR NÖRLUND AND BERNOULLI NUMBERS
If we multiply the last power series with the right-hand side of (1.2), we obtain
Ãn−k+1 µ ¶
!
∞
X n
X
¡
¢k−1
¡
s(n − j + 1, k) ¢ (j) xn
log(x + 1)
=
k! s(n − j, k) +
.
Bj
j
n−j+1
n!
j=0
n=k−1
Equating coefficients of xn in this power series and in (2.8), we get
n−k+1
X µn¶¡
¢ (j) 1
1
s(n − j, k) + n−j+1
s(n − j + 1, k) Bj = s(n, k − 1).
j
k
j=0
(3.1)
We use (2.7) in the form (n − j) s(n − j, k) + s(n − j + 1, k) = s(n − j, k − 1), add s(n − j, k)
to
¡ both sides, and divide by ¢n − j + 1; then the term in large parentheses in (3.1) becomes
s(n − j, k − 1) + s(n − j, k) /(n − j + 1), which finally gives the following result.
Theorem 3.1. For any k ≥ 1 and n ≥ k − 1 we have
n−k+1
X µn + 1¶¡
¢ (j) n + 1
s(n − j, k − 1) + s(n − j, k) Bj =
s(n, k − 1).
k
j
j=0
(3.2)
We now state two special cases, for k = 1 and k = 2. They are easily derived from (3.2)
by using (2.2)–(2.4).
Corollary 3.1. For any n ≥ 1 we have
n−1
X
j=0
(j)
Bj
(−1)j
(−1)n−1 (n)
+
Bn = 0,
(n − j)(n − j + 1) j!
n!
(3.3)
and for n ≥ 0,
n
X
(−1)j+1
j=0
(j)
Bj
Hn−j − 1
1
=
,
(n − j + 1)(n − j + 2) j!
2n + 2
(3.4)
with Hk as in (2.4).
The recurrence relation (3.3) can also be obtained independently from (1.3) by subtracting
(1.3) for n from the same identity for n − 1. Also, (3.3) was earlier obtained by Howard [5]
who used it to show that Nörlund numbers have alternating signs. The relation (3.3) is valid,
as stated, only for n ≥ 1 since s(n, 0) = δn0 ; see (2.2). Also note that H0 = 0 as an empty
sum. See Section 5 below for some further remarks on recurrences for Nörlund numbers.
4. Bernoulli Numbers of the Second Kind
We begin by stating the desired generalization of the recurrence relation (1.6).
Theorem 4.1. For any k ≥ 1 and n ≥ k we have
n−k
X
s(n − j, k)
j=0
(n − j)!
bj =
s(n − 1, k − 1)
.
(n − 1)!k
(4.1)
We state the first two cases (for k = 1, 2) separately. The next two identities follow
immediately from (4.1), with (2.3), resp. (2.4).
FEBRUARY 2010
7
THE FIBONACCI QUARTERLY
Corollary 4.1. For any n ≥ 2 we have
n−1
X
(−1)j
j=0
n−2
X
j=0
n−j
bj = 0;
(4.2)
(−1)j
1
Hn−1−j bj =
,
n−j
2(n − 1)
(4.3)
where Hk is the kth harmonic number.
The identity (4.2) is the same as (1.6), but (4.3) and the general case (4.1) appear to
be new. Theorem 4.1 would be easy to prove by considering the product of the generating
functions (1.4) and (2.8). However, it is not much more difficult to derive a substantially
more general (though complicated) result. To do this, we introduce the expression
m µ ¶
X
m s(n − r, k + m)
s̃(n, k, m) :=
,
(4.4)
r
(n − r)!
r=0
which enters through the following product of power series: With (2.8) we get
¡
¢k+m
(x + 1)m log(x + 1)
!
à ∞ µ ¶ !à ∞
X m
X (k + m)!
s(j, k + m)xj
=
xj
j!
j
j=0
j=k+m
Ãn−k−m µ ¶
!
∞
X m s(n − j, k + m)
X
= (k + m)!
xn ,
j
(n
−
j)!
j=0
n=k+m
where the inner summation can be taken to m instead of n − k − m. So with (4.4) we have
∞
X
¡
¢k+m
s̃(n, k, m)xn .
(4.5)
(x + 1)m log(x + 1)
= (k + m)!
n=k+m
We also need the following lemma, which is easy to prove by induction on m, where the
identity (2.7) will be essential.
Lemma 4.1. For any m ≥ 0 we have
m
X
dm
1
1
s(m, j)
=
(−1)j j! ¡
¢j+1 .
m
m
dx log(x + 1)
(x + 1) j=0
log(x + 1)
(4.6)
We are now ready to prove the following generalization of Theorem 4.1.
Theorem 4.2. For any m ≥ 0, k ≥ 1, and n ≥ k + m we have
n−k−m
X µm + j ¶
s̃(n − j, k, m)bm+1+j + (−1)m s̃(n + m + 1, k, m)
m
j=0
(4.7)
m
X
1
s(m, m − j)s(n, k − 1 + j)
¡k+m−1¢
,
=
(−1)m−j
n!m!(k + m) j=0
k+j−1
with s̃ as defined in (4.4).
8
VOLUME 48, NUMBER 1
RECURRENCE RELATIONS FOR NÖRLUND AND BERNOULLI NUMBERS
Proof. From (1.4) we immediately obtain, for m ≥ 0,
∞
X
dm
1
(n + m)!
m m!
= (−1) m+1 +
bn+m+1 xn .
m
dx log(x + 1)
x
n!
n=0
(4.8)
The identities (4.8) and (4.5) multiplied together give
¡
¢k+m dm
1
B(x) := (x + 1)m log(x + 1)
m
dx log(x + 1)
Ã
!
∞
n−k−m
X
X
(m + j)!
=
(k + m)!s̃(n − j, k, m)
bm+1+j xn
j!
j=0
n=k+m
+
∞
X
(−1)m m!(k + m)!s̃(n, k, m)xn−m−1
n=k+m
∞
X
=
m!(k + m)!
n=k−1
×
Ãn−k−m µ
X m + j¶
m
j=0
!
s̃(n − j, k, m)bm+1+j + (−1)m s̃(n + m + 1, k, m) xn .
(Here, as elsewhere, a sum will be empty and thus vanish when the upper limit of summation
is less than the lower limit). On the other hand, multiplying the left-hand side of (4.5) with
(4.6) we get, upon using (2.8) again,
m
X
¡
¢k+m−j−1
B(x) =
(−1)j j!s(m, j) log(x + 1)
=
j=0
m
X
j=0
m
X
¡
¢k+j−1
(−1)m−j (m − j)!s(m, m − j) log(x + 1)
∞
X
xn
n!
j=0
n=k+j−1
Ã
!
∞
m
X
X
xn
m−j
=
(−1)
(m − j)!s(m, m − j)(k + j − 1)!s(n, k + j − 1)
.
n!
j=0
n=k−1
=
(−1)
m−j
(m − j)!s(m, m − j)
(k + j − 1)!s(n, k + j − 1)
We obtain (4.7) by equating coefficients of xn in both evaluations of B(x).
¤
Note that the above proof is valid also for n < k + m; we will deal with this in the next
section.
Proof of Theorem 4.1. We set m = 0 and use the fact that from (4.4) we immediately get
s̃(n, k, 0) = s(n, k)/n!. Then we have, by (2.2),
n−k
X
s(n − j, k)
j=0
(n − j)!
bj+1 +
s(n + 1, k)
s(n, k − 1)
=
.
(n + 1)!
n!k
Finally we change the summation to range from 1 to n + 1 − k, replace n + 1 by n, and use
the fact that b0 = 1, to obtain (4.1).
¤
FEBRUARY 2010
9
THE FIBONACCI QUARTERLY
The other extreme case of Theorem 4.2 occurs when n = k+m. With (4.4) we immediately
get the following result in this case.
Corollary 4.2. For any m ≥ 0 and k ≥ 1 we have
m µ ¶
X
m s(k + m + 1 + r, k + m)
m+1
(−1)
bm+1 = (k + m)!
r
(k + m + 1 + r)!
r=0
(4.9)
m
+
X
1
s(m, m − j)s(k + m, k − 1 + j)
¡k+m−1¢
.
(−1)j+1
m!(k + m) j=0
k+j−1
It should be pointed out that this identity is far from the easiest expression of bn in terms
of Stirling numbers. In fact, the simple identity
n+1
1 X s(n, j)
bn =
n! j=1 j + 1
is proved in [6, p. 267], where this expression is also used to derive the relation
n+1
X
j=1
j!S(n, j)bj =
1
,
n+1
which involves the Stirling numbers of the second kind, in contrast to all the other identities
in this section.
5. Further Remarks
1. A notable feature of Theorem 4.2 is the fact that the identity (4.7) is a recurrence
relation of variable length. Indeed, we already considered the two extreme cases of “full
length”, namely m = 0, which led to (4.1), and “length 1”, which gave the expression (4.9).
Such formulas also exist for the classical Bernoulli numbers. They were recently studied by
the authors in [2], where a historical perspective and references are given.
Recurrence relations of variable length can also be derived for the Nörlund numbers. Given
the complexity of such formulas, we only state the following result, without proof.
For all 0 ≤ m ≤ n and all k ≥ 1 we have
n−k−m
X
j=0
m
s̃(n − j, k, m) (m+j−1)
1 X
N (n, m, k, j)
¡
¢ ,
Bm+j−1 =
(−1)j−1
j!
n! j=1
j k+m
j
where s̃ is as defined in (4.4), and
N (n, m, k, j) := ns(m − 1, j − 1)s(n − 1, k + m − j) + (m − 1)s(m − 1, j)s(n, k + m − j).
The method of proof is similar to that of Theorem 4.2. When m = 1, this reduces to (3.1),
(n)
and for the other extreme case, n = k + m, we would obtain an expression for Bn in terms
of the Stirling number of the first kind. Again, such an identity would be more complicated
than any of the simpler expressions derived by Howard [5], among them
Bn(n)
bn/2c
1 X 1
=
s(n + 1, 2k + 1),
n + 1 k=0 k + 1
which has only half the usual number of terms in the summation.
10
VOLUME 48, NUMBER 1
RECURRENCE RELATIONS FOR NÖRLUND AND BERNOULLI NUMBERS
2. If we set k = 1 and n = m in (4.7), then the summation on the left-hand side of (4.7)
vanishes. This is valid by the proof of Theorem 4.2 which holds for a wider range than just
n ≥ k + m. Thus we get
m−1
X
s(m, j)s(m, m
¡m¢
(−1)j
j
j=0
− j)
= m!(m + 1)!
m µ ¶
X
m s(2m + 1 − r, m + 1)
r=0
r
(2m + 1 − r)!
.
This is a convolution identity for Stirling numbers. Such identities have been studied before,
and will be the subject of a separate paper.
3. Two special sequences of rational numbers, namely the Cauchy numbers of the first
kind, Cn , and of the second kind, Ĉn , are closely related to the Bernoulli numbers of the
second kind and to the Nörlund numbers, respectively. They appear in Exercise 13 in [3,
p. 293] and were more recently studied in great detail in [8], but are otherwise not very well
known. While a different perspective is given in [8], using the concept of Riordan arrays,
many of the results are equivalent to those quoted in this paper, especially in Section 1. A
comparison with the defintions in [8] shows that
Cn = n!bn ,
Ĉn = Bn(n) .
Therefore our main results could also be written in terms of the Cauchy numbers of both
kinds.
Acknowledgment
We would like to thank the referee for drawing our attention to the Cauchy numbers and
to reference [8].
References
[1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards,
1964.
[2] T. Agoh and K. Dilcher, Shortened recurrence relations for Bernoulli numbers, Discrete Math., 309
(2009), 887–898.
[3] L. Comtet, Advanced Combinatorics. The Art of Finite and Infinite Expansions, Revised and enlarged
edition. D. Reidel Publ. Co., Dordrecht-Boston, 1974.
[4] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley Publ. Co.,
Reading, MA, 1989.
(n)
[5] F. T. Howard, Nörlund’s number Bn , Applications of Fibonacci Numbers, Vol. 5 (G. E. Bergum et
al., Eds.), 355–366, Kluwer Acad. Publ., Dordrecht, 1993.
[6] C. Jordan, Calculus of Finite Differences, 2nd ed., Chelsea Publ. Co., New York, 1950.
[7] D. E. Knuth, Two notes on notation, Amer. Math. Monthly, 99 (1992), 403–422.
[8] D. Merlini, R. Sprugnoli, and M. C. Verri, The Cauchy numbers, Discrete Math., 306 (2006), 1906–1920.
[9] N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924.
[10] N. J. A. Sloane, On-Line Encyclopedia of Integer Sequences,
http//www.research.att.com/~njas/sequences.
[11] E. W. Weisstein, Stirling Number of the First Kind, From MathWorld–A Wolfram Web Resource,
http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html
FEBRUARY 2010
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THE FIBONACCI QUARTERLY
MSC2000: 11B68, 05A15
Department of Mathematics, Tokyo University of Science, Noda, Chiba, 278-8510 Japan
E-mail address: agoh [email protected]
Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia,
B3H 3J5, Canada
E-mail address: [email protected]
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VOLUME 48, NUMBER 1